Difference between revisions of "Complement (linear algebra)"

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In [[linear algebra]], a '''complement''' to a subspace of a vector space is another subspace which forms an internal direct sum.  Two such spaces are mutually ''complementary''.
 
In [[linear algebra]], a '''complement''' to a subspace of a vector space is another subspace which forms an internal direct sum.  Two such spaces are mutually ''complementary''.
  
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:<math>U \cap W = \{0\} .\,</math>
 
:<math>U \cap W = \{0\} .\,</math>
  
Clearly this relation is [[symmetric]], that is, if ''W'' is a complement of ''U'' then ''U'' is also a complement of ''W''.   
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Equivalently, every element of ''V'' can be expressed uniquely as a sum of an element of ''U'' and an element of ''W''.  The complementarity relation is [[symmetric]], that is, if ''W'' is a complement of ''U'' then ''U'' is also a complement of ''W''.   
  
 
If ''V'' is finite-dimensional then for complementary subspaces ''U'', ''W'' we have
 
If ''V'' is finite-dimensional then for complementary subspaces ''U'', ''W'' we have

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In linear algebra, a complement to a subspace of a vector space is another subspace which forms an internal direct sum. Two such spaces are mutually complementary.

Formally, if U is a subspace of V, then W is a complement of U if and only if V is the direct sum of U and W, , that is:

Equivalently, every element of V can be expressed uniquely as a sum of an element of U and an element of W. The complementarity relation is symmetric, that is, if W is a complement of U then U is also a complement of W.

If V is finite-dimensional then for complementary subspaces U, W we have

In general a subspace does not have a unique complement (although the zero subspace and V itself are the unique complements each of the other). However, if V is in addition an inner product space, then there is a unique orthogonal complement